Theorem riotaprop 5996

Description: Properties of a restricted definite description operator. Todo (df-riota 5970 update): can some uses of riota2f 5993 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Hypotheses

Ref	Expression
riotaprop.0	|- F/ x ps
riotaprop.1	|- B = ( iota_ x e. A ph )
riotaprop.2	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
riotaprop	|- ( E! x e. A ph -> ( B e. A /\ ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem riotaprop

Step	Hyp	Ref	Expression
1		riotaprop.1	. . 3 |- B = ( iota_ x e. A ph )
2		riotacl 5986	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
3	1, 2	eqeltrid 2318	. 2 |- ( E! x e. A ph -> B e. A )
4	1	eqcomi 2235	. . . 4 |- ( iota_ x e. A ph ) = B
5		nfriota1 5978	. . . . . 6 |- F/_ x ( iota_ x e. A ph )
6	1, 5	nfcxfr 2371	. . . . 5 |- F/_ x B
7		riotaprop.0	. . . . 5 |- F/ x ps
8		riotaprop.2	. . . . 5 |- ( x = B -> ( ph <-> ps ) )
9	6, 7, 8	riota2f 5993	. . . 4 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
10	4, 9	mpbiri 168	. . 3 |- ( ( B e. A /\ E! x e. A ph ) -> ps )
11	3, 10	mpancom 422	. 2 |- ( E! x e. A ph -> ps )
12	3, 11	jca 306	1 |- ( E! x e. A ph -> ( B e. A /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   F/wnf 1508   e. wcel 2202   E!wreu 2512   iota_crio 5969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by:  lble  9126